Introduction to Functions

Introduction to Functions

• A function is a relationship between inputs where each input is related to exactly one output.
• Functions have a domain (set of inputs) and a codomain/range (set of possible outputs).
• Functions are typically denoted as f(x), where x represents the input.
• Examples of functions include: f(x) = sin x, f(x) = x² + 3, f(x) = 1/x, and f(x) = 2x + 3.

Types of Functions

• One-to-one function (Injective Function): Each input maps to a unique output, and no two inputs map to the same output.

• Many-to-one function: Multiple inputs can map to the same output.

• Onto function (Surjective Function): Every element in the codomain is mapped to by at least one element in the domain.

• Bijective Function: A function that is both one-to-one and onto. Invertible functions are bijective.

• Other function types mentioned are: Linear, Quadratic, Rational, Algebraic, Cubic, Modulus, Signum, Greatest Integer, Fractional Part, Even and Odd, and Periodic.

Invertible Functions

• The inverse of a function f, denoted as f⁻¹(x), reverses the relationship, mapping outputs back to inputs.
• A function is invertible if and only if it is bijective (one-to-one and onto).
• For a function to have an inverse, each output must correspond to a unique input.
• The inverse of adding is subtracting, and the inverse of multiplying is dividing.

Graphical Representation

• The domain of a function is the set of inputs, and the range is the set of outputs.
• The domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x).
• Visual representation aids understanding of function relationships and inverses.

Reflexive Property of Invertible Functions

• If a point (a, b) is on the graph of a function f, then the point (b, a) must be on the graph of the inverse function f⁻¹.
• This property demonstrates the reflection of the graph of a function over the line y = x when finding the inverse.

Conditions for Invertibility

• A function is invertible if it's both one-to-one and onto over its entire domain.
• A function may be one-to-one in certain parts of its domain but not over the entire domain.

Horizontal Line Test

• Used to determine if a function is invertible (one-to-one).
• If any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.

Demonstration of an Invertible Function

• Graphing y = 2x and y = x/2, illustrates the relationship between a function and its inverse.
• The inverse graph is a reflection of the original graph across the line y = x.

Properties of the Inverse Graph

• The inverse of a function is found by exchanging x and y coordinates in its graph.
• Graph of the inverse of f is a reflection across the y = x line.
• Example in page 9 details coordinates shifting in this relation.

Conclusion regarding invertibility

• Functions must be one-to-one to have an inverse.
• Exchanging input and output rows is how to find the inverse of tabular functions.
• Many "toolkit functions" have inverses.
• Graphically, an inverse is a reflection across the line y = x.